Since \(KG(a,1)\)  is the complete graph on \(a\)  vertices and \(\beta _k(K_a)=k\)  for \(1\le k\le a\) , all the possible values of \(\beta _k\big (KG(a,1)\big )\)  are known. In 1959, Erdős and Gallai [1]} proved that Conjecture  is true when \(b = 2\) , which justifies equality in the cases considered in Example . In 2016, Frankl [2]} proved that the conjecture is true when \(b=3\) . For more recent results and bounds regarding this conjecture we suggest [3]} and [4]}. We can now present the upper bound for the fractional \(\mathbb {K}_k\) -cover number of \(a\!\!:\!\!b\) -colourable graphs.
